The Orthogonal Procrustes Problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices, $A$ and $B$, and is asked to find an orthogonal matrix $R$ that most closely maps $A$ to $B$.
Questions tagged [procrustes-problem]
27 questions
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Showing that matrix $Q=UV^T$ is the nearest orthogonal matrix to $A$.
Let $A$ be an $m \times n$ matrix with a singular value decomposition $A=U\Sigma V^T$. Show that the matrix $Q=UV^T$ is the nearest orthogonal matrix to $A$, i.e.,
$$\min_{Q^TQ=I_{n \times n}} \|A-Q\|_F$$
BadAtMath
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Solve least-squares minimization from overdetermined system with orthonormal constraint
I would like to find the rectangular matrix $X \in \mathbb{R}^{n \times k}$ that solves the following minimization problem:
$$
\mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times k}} \left\| A X - B \right\|_F^2 \quad \text{ subject to } X^T X =…
Alec Jacobson
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Find a permutation of the rows of a matrix that minimizes the sum of squared errors
I'm struggling with the following problem:
Let $A, B \in \mathbb R^{n \times d}$. Denote by $\mathcal{P}$ the set of all possible permutations of the rows of $A$. Find a permutation $\pi \in \mathcal{P}$ that minimizes $$\sum_{i=1}^n\sum_{j=1}^d…
Eva
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Orthogonal Procrustes Variant
(author note: this question was also asked on mathoverflow).
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$…
Matt
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Orthogonal Procrustes Problem using the operator norm
If $A, B \in \mathbb{R}^{n \times r}$ are two matrices, the solution to the so-called Orthogonal Procrustes Problem
$$\min_{O^TO=I_r} \|AO-B\|$$
is given by the polar factor of $A^TB$ whenever the norm is the Frobenius norm. The minimization here…
squattyroo
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Orthogonal Procrustes Problem
The classical Orthogonal Procrustes Problem is
$$\begin{array}{ll} \text{minimize} & \|A\Omega-B\|_{F}\\ \text{subject to} & \Omega'\Omega=I\end{array}$$
where $A$ and $B$ are known matrices.
Suppose $A$ is the identity matrix. I would like to…
Lindon
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Solution to a Procrustes-like Problem
I recently came across the following problem that resembles a Procrustes problem and I wonder if an analytic solution for this problem might exist:
$$\underset{(R,\alpha)}{\operatorname{argmin}} ||RAe^{j\alpha W}-B||_F$$
Where $A,B \in \mathbb{C}^{3…
Mantabit
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Optimizing Trace$(Q^TZ)$ subject to $Q^TQ=I$
Let $Z \in \mathbb{R}^{m \times n}$ be a tall matrix ($m > n$). Solve the following optimization problem in $Q \in \mathbb{R}^{m \times n}$
$$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(Q^T Z \right)\\ \text{subject to} & Q^T Q = I_{n…
kkcocoqq
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Convexity of the orthogonal Procrustes problem
Given two orthogonal matrices ${\bf{M}} \in {{\Bbb{R}}^{m \times n}}$ and ${\bf{N}} \in {{\Bbb{R}}^{m \times n}}$, there is an orthogonal transformation matrix ${\bf{T}} \in {{\Bbb{R}}^{n \times n}}$ which closely maps ${\bf{M}}$ and…
ar_k
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Orthogonal (unitary) Procrustes problem (complex matrices)
The orthogonal Procrustes problem can be stated as finding the orthogonal matrix $\Omega$ that maps $A$ most closely to $B$
$$\arg\min_{\Omega}\|A\Omega - B\|_F \quad\mathrm{subject\ to}\quad \Omega^T
\Omega=I$$
The solution is well known and found…
zilver
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Nearest (in the Frobenius sense) semi-orthogonal matrix
I found a question about finding a nearest semi-orthogonal matrix, but I need to find the nearest semi-orthogonal matrix subject to a slightly different constraint. Given $m \times n$ matrix $M$,
$$ \begin{array}{ll} \underset {R \in \Bbb R^{m…
Buu Pham
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Find an orthogonal matrix to minimize the norm
Let $A\in \mathbb{R}^{n\times n}$. Find $\overline O$, orthogonal matrix, to minimize $\|A-O\|_F$. That is;
$$\min_{O\in O(n)} \|A-O\|_F$$
Where $O(n)$ are the set of orthogonal matrices of $\mathbb{R}^{n\times n}$.
$\|\cdot \|_F $ is the Frobenius…
blueplusgreen
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Variant of orthogonal Procrustes problem
I want to find an anti-symmetric matrix $T$ which minimizes $\|A-e^TBe^{-T}\|^2 + \mu\|T\|^2$, where $A$ and $B$ are symmetric positive definite matrices and the norm is the Frobenius matrix norm. The original orthogonal Procrustes problem is to…
uekstrom
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Find $Q$ in $R_2 = Q^TR_1Q$ where all matrices $R_1, R_2, Q$ are 3x3 rotations
Can we obtain $Q$ in
$$R_2 = Q^TR_1Q$$
where $R_1, R_2, Q$ are all 3x3 rotation matrices, $R_1 \ne R2$, and $\operatorname{trace}(R_1)=\operatorname{trace}(R_2)$?
I have a problem where $R_1$ and $R_2$ are known. I also have the ground truth of $Q$,…
Kay
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How is Procrustes Distance Defined?
How is Procrustes Distance defined between two given datasets?
Assume that the datasets each of them having "k" points in "n" dimensions.
I couldn't find a proper source anywhere with a mathematical expression. I found some leads, but they are in…
truth_seeker
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